Introduction to Topology of Real Algebraic Varieties
1. The Early Topological Study of Real Algebraic Plane Curves
1.1. Basic Definitions and Problems
A curve (at least, an algebraic curve) is something more than just the set of points which belong to it. There are many ways to introduce algebraic curves. In the elementary situation of real plane projective curves the simplest and most convenient is the following definition, which at first glance seems to be overly algebraic.
By a real projective algebraic plane curve11Of course,
the full designation is used only in formal
situations. One normally adopts an abbreviated terminology. We shall
say simply a curve in contexts where this will not lead to
confusion. of degree
A point
In the topology of nonsingular real projective algebraic plane curves, as in other similar areas, the first natural questions that arise are classification problems.
1.1.A Topological Classification Problem.
Up to homeomorphism, what are the possible
sets of real points of a nonsingular real projective algebraic plane
curve of degree
1.1.B Isotopy Classification Problem.
Up
to homeomorphism, what are the possible pairs
It is well known that the components of a closed one-dimensional manifold are
homeomorphic to a circle, and the topological type of the manifold is
determined by the number of components; thus, the first problem reduces to
asking about the number of components of a curve of degree
The second problem has a more naive formulation as the question of how a
nonsingular curve of degree
1.2. Digression: the Topology of Closed One-Dimensional Submanifolds of the Projective Plane
For brevity, we shall refer to closed one-dimensional submanifolds of the projective plane as topological plane curves, or simply curves when there is no danger of confusion.
A connected curve can be situated in
Any two one-sided connected curves intersect, since each of them
realizes the nonzero element of the group
Two disjoint ovals can be situated in two topologically distinct ways: each may lie outside the other one—i.e., each is in the outside component of the complement of the other—or else they may form an injective pair, i.e., one of them is in the inside component of the complement of the other—in that case, we say that the first is the inner oval of the pair and the second is the outer oval. In the latter case we also say that the outer oval of the pair envelopes the inner oval.
A set of
The pair
When depicting a topological plane curve one usually represents the projective
plane either as a disc with opposite points of the boundary identified, or else
as the compactification of
1.3. Bézout’s Prohibitions and the Harnack Inequality
The most elementary prohibitions, it seems, are the topological consequences of Bézout’s theorem. In any case, these were the first prohibitions to be discovered.
1.3.A Bézout’s Theorem (see, for example, Wal-50, Sha-77).
Let
1.3.B Corollary (1).
A nonsingular plane curve of degree
In fact, in order for a nonsingular plane curve to be two-sided, i.e., to be
homologous to zero
1.3.C Corollary (2).
The number of ovals in the union of two
nests of a nonsingular plane curve of degree
To prove Corollary 2 it suffices to apply Bézout’s theorem to the curve and
to a line which passes through the insides of the smallest ovals in the
nests.
1.3.D Corollary (3).
There can be no more than
To prove Corollary 3 it suffices to apply Bézout’s theorem to the curve and
to a conic which passes through the insides of the smallest ovals in the
nests.
One can give corollaries whose proofs use curves of higher degree than lines and conics (see Section 3.8). The most important of such results is Harnack’s inequality.
1.3.E Corollary (4 (Harnack Inequality Har-76)).
The number of components of a
nonsingular plane curve of degree
The derivation of Harnack Inequality from Bézout’s theorem can be found in Har-76, and also Gud-74. However, it is possible to prove Harnack Inequality without using Bézout’s theorem; see, for example, Gud-74, Wil-78 and Section 3.2 below.
1.4. Curves of Degree ≤5
If
For
1.4.A
Isotopy Classification of Nonsingular
Real Plane Projective Curves of Degree
The curves of degree
It will be left to the reader to prove that one in fact obtains the curves in Figure 2 as a result; alternatively, the reader can deduce this fact from the theorem in the next subsection.
The isotopy types of nonempty nonsingular curves of degree 4 can be realized in
a similar way by small perturbations of a union of two conics which intersect
in four real points (Figure 3). An empty curve of degree 4
can be defined, for example, by the equation
All of the isotopy types of nonsingular curves of degree 5 can be realized by
small perturbations of the union of two conics and a line, shown in
Figure 4.
For the isotopy classification of nonsingular curves of degree 6 it is no longer sufficient to use this type of construction, or even the prohibitions in the previous subsection. See Section 1.13.
1.5. The Classical Method of Constructing Nonsingular Plane Curves
All of the classical constructions of the topology of nonsingular plane curves are based on a single construction, which I will call classical small perturbation. Some special cases were given in the previous subsection. Here I will give a detailed description of the conditions under which it can be applied and the results.
We say that a real singular point
1.5.A Classical Small Perturbation Theorem (see Figure 5).
Let
Then there exists a nonsingular plane curve
(1)
(2) For each component
(3)
(4)
(5)
(6) If
There exists
It follows from (1)–(3) that for fixed
We say that the curves defined by the polynomials
Proof.
Proof of Theorem 1.5.A
We set
1.6. Harnack Curves
In 1876, Harnack Har-76 not only proved the inequality 1.3.E in Section 1.3, but also completed the topological classification of nonsingular plane curves by proving the following theorem.
1.6.A Harnack Theorem.
For any natural number
| (1) |
there exists a nonsingular plane curve of degree
The inequality on the right in 1 is Harnack Inequality. The
inequality on the left is part of Corollary 1 of Bézout’s theorem
(see Section 1.3.B). Thus, Harnack Theorem together with
theorems 1.3.B and 1.3.E actually give a complete
characterization of the set of topological types of nonsingular plane
curves of degree
Curves with the maximum number of components (i.e., with
1.6.B .
For any natural number
Proof.
We shall actually construct a sequence of M-curves. At each
step of the construction we add a line to the M-curve just
constructed, and then give a slight perturbation to the union. We can
begin the construction with a line or, as in Harnack’s proof in
Har-76, with a circle. However, since we have already treated
curves of degree
Recall that we obtained a degree 5 M-curve by perturbing the union of two
conics and a line
We now construct a sequence of auxiliary curves
We construct the M-curve
are only slightly deformed—so that the
number of components of
The proof that the left inequality in 1 is best possible, i.e.,
that there is
a curve with the minimum number of components, is much simpler. For example,
we can take the curve given by the equation
By choosing the auxiliary curves
1.7. Digression: the Space of Real Projective Plane Curves
By the definition of real projective algebraic plane curves of
degree
We let
Curves which belong to the same component of the space
1.7.A Rigid Isotopy Classification Problem.
Classify the nonsingularcurves of degree
If
Although this section is devoted to the early stages of the theory, I cannot
resist commenting in some detail about a more recent result. In 1978,
V. A. Rokhlin Rok-78 discovered that for
We now examine the subset of
It is clear that a curve of degree
We now consider the space
Using an argument similar to the proof that
There are two types of nondegenerate real points on a plane curve. We say that
a nondegenerate real double point
If a curve of degree
A line in
By the transversality theorem, the pencils which intersect the set of real
singular curves only at points of the principal part and only transversally
form an open everywhere dense subset of the set of all real pencils of curves
of degree
1.8. End of the Proof of Theorem 1.6.A
In Section 1.6 it was shown that for any
1.9. Isotopy Types of Harnack M-Curves
Harnack’s construction of M-curves in Har-76
differs from the construction in the proof of Theorem 1.6.B in
that a conic, rather than a curve of degree 5, is used as the original
curve. Figure 9 shows that the M-curves
of degree
In these constructions one obtains different isotopy types of M-curves
depending on the choice of auxiliary curves (more precisely, depending on the
relative location of the intersections
In conclusion, we mention two curious properties of Harnack M-curves, for which the reader can easily furnish a proof.
1.9.A .
The depth of a nest in a Harnack M-curve is at most 2.
1.9.B .
Any Harnack M-curve of even degree
1.10. Hilbert Curves
In 1891 Hilbert Hil-91 seems to have been the first to clearly state the isotopy classification problem for nonsingular curves. As we saw, the isotopy types of Harnack M-curves are very special. Hilbert suggested that from the topological viewpoints M-curves are the most interesting. This Hilbert’s guess was strongly confirmed by the whole subsequent development of the field.
There is a big gap between property 1.9.A of Harnack M-curves and the corresponding prohibition in 1.3.C. Hilbert Hil-91 showed that this gap is explained by the peculiarities of the construction and not by the intrinsic properties of M-curves. He proposed a new method of constructing M-curves which was close to Harnack’s method, but which gives M-curves with nests of any depth allowed by Theorem 1.3.C. In his method the role a line plays in Harnack’s method is played instead by a nonsingular conic, and a line or a conic is used for the starting curve. Figures 10–11 show how to construct M-curves by Hilbert’s method.
In Table 3 we list the isotopy types of M-curves of degree
The first difficult special problems that Hilbert met were related with
curves of degree 6. Hilbert succeeded to construct M-curves of degree
In fact, Hilbert invented a method which allows to answer to all questions on topology of curves of degree 6. It involves a detailed analysis of singular curves which could be obtained from a given nonsingular one. The method required complicated fragments of singularity theory, which had not been elaborated at the time of Hilbert. Completely this project was realized only in the sixties by D. A. Gudkov. It was Gudkov who obtained a complete table of real schemes of curves of degree 6.
Coming back to Hilbert, we have to mention his famous problem list Hil-01. He included into the list, as a part of the sixteenth problem, a general question on topology of real algebraic varieties and more special questions like the problem on mutual position of components of a plane curve of degree 6.
The most mysterious in this problem seems to be its number. The number sixteen plays a very special role in topology of real algebraic varieties. It is difficult to believe that Hilbert was aware of that. It became clear only in the beginning of seventies (see Rokhlin’s paper “Congruences modulo sixteen in the sixteenth Hilbert’s problem” Rok-72). Nonetheless, sixteen was the number assigned by Hilbert to the problem.
1.11. Analysis of the Results of the Constructions. Ragsdale
In 1906, V. Ragsdale Rag-06 made a remarkable
attempt to guess new prohibitions, based on the results of the
constructions by Harnack’s and Hilbert’s methods. She concentrated her
attention on the case of curves of even degree, motivated by the
following special properties of such curves. Since a curve of an even
degree is two-sided, it divides
By analyzing the constructions, Ragsdale Rag-06 made the following observations.
1.11.A (compare with 1.9.A and 1.9.B).
For any Harnack M-curve of even degree
1.11.B .
For any Hilbert M-curve of even
degree
This gave her evidence for the following conjecture.
1.11.C Ragsdale Conjecture.
For any curve of even degree
The most mysterious in this problem seems to be its number. The number sixteen plays a very special role in topology of real algebraic varieties. It is difficult to believe that Hilbert was aware of that. It became clear only in the beginning of seventies (see Rokhlin’s paper “Congruences modulo sixteen in the sixteenth Hilbert’s problem” Rok-72). Nonetheless, sixteen was the number assigned by Hilbert to the problem.
Writing cautiously, Ragsdale formulated also weaker conjectures. About thirty years later I. G. Petrovsky Pet-33, Pet-38 proved one of these weaker conjectures. See below Subsection 1.13.
Petrovsky also formulated conjectures about the upper bounds for
Both Ragsdale Conjecture formulated above and its version stated by
Petrovsky Pet-38 are wrong. However they stayed for rather long
time: Ragsdale Conjecture on
In Section LABEL:s5.4s we shall return to this very first conjecture of a general nature on the topology of real algebraic curves. At this point we shall only mention that several weaker assertions have been proved and examples have been constructed which made it necessary to weaken the second inequality by 1. In the weaker form the Ragsdale conjecture has not yet been either proved or disproved.
The numbers
Maybe mathematicians trying to conjecture restrictions on some integer should keep this case in mind as an evidence that restrictions can have not only the shape of inequality, but congruence. Proof of these Gudkov’s conjectures initiated by Arnold Arn-71 and completed by Rokhlin Rok-72, Kharlamov Kha-73, Gudkov and Krakhnov GK-73 had marked the beginning of the most recent stage in the development of the topology of real algebraic curves. We shall come to this story at the end of this Section.
1.12. Generalizations of Harnack’s and Hilbert’s Methods. Brusotti. Wiman
Ragsdale’s work Rag-06 was partly
inspired by the erroneous paper of Hulbrut, containing a proof of the false assertion that an M-curve
can be obtained by means
of a classical small perturbation (see Section 1.5) from only
two M-curves, one of which must have degree
In 1910–1917, L. Brusotti showed that this is not the case. He found inductive constructions of M-curves based on classical small perturbation which were different from the methods of Harnack and Hilbert.
Before describing Brusotti’s constructions, we need some definitions. A simple
arc
An M-curve
a) The intersection
b) The cyclic orders determined on the intersection
c)
d) If
e) The rank of the base
An auxiliary curve can be the empty curve of degree 0. In this case the rank
of
Let
Any simple arc of a curve of degree
If the generating curve has degree 1 and the auxiliary curve has degree 2, then the Brusotti construction turns out to be Harnack’s construction. The same happens if we take an auxiliary curve of degree 1 or 0. If the generating curve has degree 2 and the auxiliary curve has degree 1 or 2 (or 0), then the Brusotti construction is the same as Hilbert’s construction.
In general, not all Harnack and Hilbert constructions are included in
Brusotti’s scheme; however, the Brusotti construction can easily be extended in
such a way as to be a true generalization of the Harnack and Hilbert
constructions. This extension involves allowing the use of an arbitrary number
of bases of the generating curve. Such an extension is particularly worthwhile
when the generating curve has degree
It can be shown that Brusotti’s construction with generating curve of degree 1
and auxiliary curve of degree
Even with the first stage of Brusotti’s construction, i.e., the classical small
perturbation of the union of the curve and the line, one obtains an M-curve
(of degree 6) which has isotopy type
In Figures 13 and 14 we show the construction of two curves of degree 6 which are auxiliary curves with respect to a line. In this case the Brusotti construction gives new isotopy types beginning with degree 8.
In the Hilbert construction we keep track of the location relative to a fixed
line
In numerous papers by Brusotti and his students, many series of Brusotti M-curves were found. Generally, new isotopy types appear in them beginning with degree 9 or 10. In these constructions they paid much attention to combinations of nests of different depths—a theme which no longer seems to be very interesting. An idea of the nature of the results in these papers can be obtained from Gudkov’s survey Gud-74; for more details, see Brusotti’s survey Bru-56 and the papers cited there.
An important variant of the classical constructions of M-curves, of which we
shall need to make use in the next section, is not subsumed under Brusotti’s
scheme even in its extended form. This variant, proposed by Wiman
Wim-23, consists in the following. We take an M-curve
1.13. The First Prohibitions not Obtained from Bézout’s Theorem
The techniques discussed above
are, in essence, completely elementary. As we saw (Section
1.4), they are sufficient to solve the isotopy classification
problem for nonsingular projective curves of degree
1.13.A Petrovsky Theorem (Pet-33, Pet-38).
For any nonsingular real projective
algebraic plane curve of degree
| (2) |
(Recall that
As it follows from Pet-33 and Pet-38, Petrovsky did not know Ragsdale’s paper. But his proof runs along the lines indicated by Ragsdale. He also reduced the problem to estimates of Euler characteristic of the pencil curves, but he went further: he proved these estimates. Petrovsky’s proof was based on a technique that was new in the study of the topology of real curves: the Euler-Jacobi interpolation formula. Petrovsky’s theorem was generalized by Petrovsky and Oleinik PO-49 to the case of varieties of arbitrary dimension, and by Oleĭnik Ole-51 to the case of curves on a surface. More about the proof and the influence of Petrovsky’s work on the subsequent development of the subject can be found in Kharlamov’s survey Kha-86 in Petrovsky’s collected works. I will only comment that in application to nonsingular projective plane curves, the full potential of Petrovsky’s method, insofar as we are able to judge, was immediately realized by Petrovsky himself.
We now turn to Gudkov’s work. In a series of papers in the 1950’s and 1960’s, he completed the development of the techniques needed to realize Hilbert’s approach to the problem of classifying curves of degree 6 (these techniques were referred to as the Hilbert-Rohn method by Gudkov), and he used the techniques to solve this problem (see GU-69). The answer turned out to be elegant and stimulating.
1.13.B Gudkov’s Theorem GU-69.
The 56 isotopy types listed in Table 4, and no others, can be realized by nonsingular real projective algebraic plane curves of degree 6.
This result, along with the available examples of curves of higher degree, led Gudkov to the following conjectures.
1.13.C Gudkov Conjectures GU-69.
(i) For any
M-curve of even degree
(ii) For any
While attempting to prove conjecture 1.13.C(i), V. I. Arnold Arn-71 discovered some striking connections between the topology of a real algebraic plane curve and the topology of its complexification. Although he was able to prove the conjecture itself only in a weaker form (modulo 4 rather than 8), the new point of view he introduced to the subject opened up a remarkable perspective, and in fact immediately brought fruit: in the same paper Arn-71 Arnold proved several new prohibitions (in particular, he strengthened Petrovsky’s inequalities 1.13.A). The full conjecture 1.13.C(i) and its high-dimensional generalizations were proved by Rokhlin Rok-72, based on the connections discovered by Arnold in Arn-71.
I am recounting this story briefly here only to finish the preliminary history exposition. At this point the technique aspects are getting too complicated for a light exposition. After all, the prohibitions, which were the main contents of the development at the time we come to, are not the main subject of this book. Therefore I want to switch to more selective exposition emphasizing the most profound ideas rather than historical sequence of results.
A reader who prefare historic exposition can find it in Gudkov’s survey article Gud-74. To learn about the many results obtained using methods from the modern topology of manifolds and complex algebraic geometry (the use of which was begun by Arnold in Arn-71), the reader is referred to the surveys Wil-78, Rok-78, Arn-79, Kha-78, Kha-86, Vir-86.
Exercises
1.1 What is the maximal number
1.2 Prove the Harnack inequality (the right hand side of (1)) deducing it from the Bézout Theorem.
2. A Real Algebraic Curve from the Complex Point of View
2.1. Complex Topological Characteristics of a Real Curve
According to a tradition going back to Hilbert, for
a long time the main question concerning the topology of real
algebraic curves was considered to be the determination of which
isotopy types are realized by nonsingular real projective algebraic
plane curves of a given degree (i.e., Problem 1.1.B above).
However, as early as in 1876 F. Klein Kle-22 posed the
question more broadly. He was also interested in how the isotopy type
of a curve is connected to the way the set
The set
The complex analogue of Isotopy Classification Problem 1.1.B
leads also to a trivial classification: the topology of the pair
The set
The real curve
A pair of orientations opposite to each other is called a semiorientation. Thus the complex orientations of a curve of type I comprise a semiorientation. Naturally, the latter is called a complex semiorientation.
The scheme of relative location of the ovals of a curve is called the real scheme of the curve. The real scheme enhanced by the type of the curve, and, in the case of type I, also by the complex orientations, is called the complex scheme of the curve.
We say that the real scheme of a curve of degree
The division of curves into types is due to Klein Kle-22. It was Rokhlin Rok-74 who introduced the complex orientations. He introduced also the notion of complex scheme and its type Rok-78. In the eighties the point of view on the problems in the topology of real algebraic varieties was broadened so that the role of the main object passed from the set of real points, to this set together with its position in the complexification. This viewpoint was also promoted by Rokhlin.
As we will see, the notion of complex scheme is useful even from the point of view of purely real problems. In particular, the complex scheme of a curve is preserved under a rigid isotopy. Therefore if two curves have the same real scheme, but distinct complex schemes, the curves are not rigidly isotopic. The simplest example of this sort is provided by the curves of degree 5 shown in Figure 8, which are isotopic but not rigidly isotopic.
2.2. The First Examples
A complex projective line is
homeomorphic to the two-dimensional sphere.44I believe
that this may be assumed well-known. A short explanation
is that a projective line is a one-point compactification of an
affine line, which, in the complex case, is homeomorphic to
The action of
The set of complex points of a nonsingular plane projective conic is
homeomorphic to
2.3. Classical Small Perturbations from the Complex Point of View
To consider further examples, it would be useful to understand what is going on in the complex domain, when one makes a classical small perturbation (see Section 1.5).
First, consider the simplest special case: a small perturbation
of the union of two real lines. Denote the lines by
This is the complex view of the picture. Up to this point it does not matter whether the curves are defined by real equations or not.
To relate this to the real view presented in Section 1.5, one
needs to describe the position of the real parts of the curves in
their complexifications and the action of
This is almost complete description. It misses only one point: one has to specify which half-discs are connected with each other by a half-annulus.
First, observe, that the halves of the complex point set of any curve of type I can be distinguished by the orientations of the real part. Each of the halves has the canonical orientation defined by the complex structure, and this orientation induces an orientation on the boundary of the half. This is one of the complex orientations. The other complex orientation comes from the other half. Hence the halves of the complexification are in one-to-one correspondence to the complex orientations.
Now we have an easy answer to the question above.
The halves of
The union of two lines can be perturbed in two different ways. On the other hand, there are two ways to connect the halves of their complexifications. It is easy to see that different connections correspond to different perturbations. See Figure 16.
The special classical small perturbation considered above is a key for
understanding what happens in the complex domain at an arbitrary
classical small perturbation. First, look at the complex picture,
forgetting about the real part. Take a plane projective curve, which
has only nondegenerate double points. Near such a point it is organized
as a union of two lines intersecting at the point. This means that there
are a neighborhood
For example, take the union of
By the way, this description shows that the complex point set of a
nonsingular plane projective curve of degree
Now let us try to figure out what happens with the complex schemes in an arbitrary classical small perturbation of real algebraic curves. The general case requirs some technique. Therefore we restrict ourselves to the following intermediate assertion.
2.3.A .
(Fiedler Rok-78, Section 3.7
and Marin Mar-80.) Let
If it takes place, then the orientation of
Proof.
If some of
Assume now that all
Again assume that all
2.4. Further Examples
Although Theorem 2.3.A describes only a very special class of classical small perturbations (namely perturbations of unions of nonsingular curves intersecting only in real points), it is enough for all constructions considered in Section 1. In Figures 17, 18, 19, 20, 21, 22 and 23 I reproduce the constructions of Figures 2, 3, 4, 6, 7, 10 and 11, enhancing them with complex orientations if the curve is of type I.
2.5. Digression: Oriented Topological Plane Curves
Consider an oriented topological plane curve, i. e. an oriented closed one-dimensional submanifold of the projective plane, cf. 1.2.
A pair of its ovals is said to be injective if one of the ovals is enveloped by the other.
An injective pair of ovals is said
to be positive if the orientations of the ovals determined by
the orientation of the entire curve are induced by an orientation of
the annulus bounded by the ovals. Otherwise, the injective pair of
ovals is said to be negative. See Figure 24.
It is clear that the division of
pairs of ovals into positive and negative pairs does not change if the
orientation of the entire curve is reversed; thus, the injective pairs
of ovals of a semioriented curve (and, in particular, a curve of type I)
are divided into positive and negative.
We let
The ovals of an
oriented curve one-sidedly embedded into
To describe a semioriented topological plane curve (up to
homeomorphism of the projective plane) we need to enhance the coding
system introduced in 1.2. The symbols representing positive
ovals will be equipped with a superscript
To describe the complex scheme of a curve of degree
It is easy to check, that the coding of this kind of the complex scheme
of a plane projective real algebraic curve describes the union of
In these notations, the complex schemes of cubic curves shown in Figure
17 are
The complex schemes of quartic curves realized in
Figure 18 are
The complex schemes of quintic curves realized in
Figure 19 are
In fact, these lists of complex schemes contain all schemes of nonsingular algebraic curves for degrees 3 and 5 and all nonempty schemes for degree 4. To prove this, we need not only constructions, but also restrictions on complex schemes. In the next two sections restrictions sufficient for this will be provided.
2.6. The Simplest Restrictions on a Complex Scheme
To begin with, recall the following obvious restriction, which was used in Section 2.2.
2.6.A .
A curve with empty real point set is of type
II.
The next theorem is in a sense dual to 2.6.A.
2.6.B .
An M-curve is of type I.
Proof.
Let
disjoint circles
lying on
Then cap each
boundary circle with a disk. Each component of
2.6.C Klein’s Congruence (see Kle-22, page 172).
If
Proof.
Consider a half of
2.6.D A Nest of the Maximal Depth (see Rok-78, 3.6).
A real scheme of degree
Such a scheme exists and is unique for any
I preface the proof of 2.6.D with a construction interesting for
its own. It provides a kind of window through which one can take a look
at the imaginary part of
As we know (see Section 2.2), the complex point set of a real
line is divided by its real point set into two halves, which are in a
natural one-to-one correspondence with the orientations of the real
line. The set of all real lines on the projective plane is the real
point set of the dual projective plane. The halves of lines comprise a
two-dimensional sphere covering this projective plane. An especially
clear picture of these identifications appears, if one identifies real
lines on the projective plane with real planes in
There is a unique real line passing through any imaginary point of
Consequently, there is a unique half of a real line containing an
imaginary point of
Proof of 2.6.D.
Let
2.7. Rokhlin’s Complex Orientation Formula
Now we shall consider a powerful restriction on a complex orientation of a curve of type I. It is powerful enough to imply restrictions even on real schemes of type I. The first version of this restriction was published in 1974, see Rok-74. There Rokhlin considered only the case of an algebraic M-curve of even degree. In Mis-75 Mishachev considered the case of an algebraic M-curve of odd degree. For an arbitrary nonsingular algebraic curve of type I, it was formulated by Rokhlin Rok-78 in 1978. The proofs from Rok-74 and Mis-75 work in this general case. The only reason to restrict the main formulations in these early papers to M-curves was the traditional viewpoint on the subject of the topology of real plane algebraic curves.
Here are Rokhlin’s formulations from Rok-78.
2.7.A Rokhlin Formula.
If the degree
2.7.B Rokhlin-Mishachev Formula.
If
Theorems 2.7.A and 2.7.B can be united into a single formulation. This requires, however, two preliminary definitions.
First, given an oriented topological curve
The second prerequisite notion is a sort of unusual integration:
an integration with respect to the
Euler characteristic, in which the Euler
characteristic plays the role of a measure. It is well known that the
Euler characteristic shares an important property of measures: it is
additive in the sense that for any sets
However, the Euler characteristic is neither
For details and applications of that notion, see Vir-88.
Now we can unite 2.7.A and 2.7.B:
2.7.C Rokhlin Complex Orientation Formula.
If
Here I give a proof of 2.7.C, skipping the most
complicated details. Take a curve
Now take the cycle
First, it is easy to see
that the homology class
Second, one may calculate the same intersection number geometrically:
moving the cycles into a general position and counting the local
intersection numbers. I will perturb the cycle
I omit the proof of the latter statement. It is nothing but a
straightforward checking that multiplication by
Now recall that the sum of indices of a vector field tangent to the
boundary of a compact manifold is equal to the Euler characteristic of
the manifold. Therefore the input of singular points lying in a
connected component of
2.7.D Corollary 1. Arnold Congruence.
For a
curve of an even degree
Proof.
Observe that in the case of an even degree
Thus
Recall that
Denote the number of all injective pairs of ovals for a curve under
consideration by
2.7.E Corollary 2.
For any curve of an even degree
Proof.
By 2.7.A
2.7.F Corollary 3.
For any curve of an odd degree
Proof.
Since
On the other hand,
2.8. Complex Schemes of Degree ≤5
As it was promised in Section 2.5, we can prove now that only schemes realized in Figures 17, 18 and 19 are realizable by curves of degree 3, 4 and 5, respectively. For reader’s convinience, I present here a list of all these complex schemes in Table 5.
Degree 3. By Harnack’s inequality, the number of components is at
most 2. By 1.3.B a curve of degree 3 is one-sided, thereby the
number of components is at least 1. In the case of 1
component the real scheme
is
Degree 4. By Harnack’s inequality the number of
components is at most 4. We know (see 1.4) that only real
schemes
The scheme
By 2.6.D the scheme
Degree 5. By Harnack’s inequality the number of
components is at most 7. We know (see 1.4) that only real
schemes
By 2.6.B
By 2.6.D
The real scheme
Exercises
2.1 Prove that for
any two semioriented curves with the same code (of the kind introduced
in 3.7) there exists a homeomorphism of
2.2 Prove that for any two curves
2.3 Deduce 2.7.A and 2.7.B from 2.7.C and, vise versa, 2.7.C from 2.7.A and 2.7.B.
3. The Topological Point of View on Prohibitions
3.1. Flexible Curves
In Section 1 all prohibitions were deduced from the Bézout
Theorem. In Section 2 many proofs were purely topological. A
straightforward analysis shows that the proofs of all prohibitions are
based on a small number of basic properties of the complexification of
a nonsingular plane projective algebraic curve. It is not difficult to
list all these properties of such a curve
-
(1)
Bézout’s theorem;
-
(2)
ℂA realizes the classm[ℂP1]∈H2(ℂP2) ; -
(3)
ℂA is homeomorphic to a sphere with(m−1)(m−2)/2 handles; -
(4)
conj(ℂA)=conj ; -
(5)
the tangent plane to
ℂA at a pointx∈ℝA is the complexification of the tangent line ofℝA atx .
The last four are rough topological properties. Bézout’s theorem
occupies a special position. If we assume that some surface
smoothly embedded into
Therefore, along with algebraic curves, it is useful to consider objects which imitate them topologically.
An oriented smooth closed connected two-dimensional submanifold
-
(i)ii
S realizesm[ℂP1]∈H2(ℂP2) ; -
(ii)i
the genus of
S is equal to(m−1)(m−2)/2 ; -
(iii)
conj(S)=S ; -
(iv)
the field of planes tangent to
S onS∩ℝP2 can be deformed in the class of planes invariant underconj into the field of (complex) lines inℂP2 which are tangent toS∩ℝP2 .
A flexible curve
3.2. The Most Elementary Prohibitions on Real Topology of a Flexible Curve
The simplest prohibitions
are not related to the position of
The most important of these prohibitions is Harnack’s inequality. Recall that it is
where
3.2.A .
For a
reversing orientation involution
In turn, 3.2.A can be deduced from the following purely topological theorem on involutions:
3.2.B Smith-Floyd Theorem.
For any involution
This theorem is one of the most famous results of the Smith theory. It is deduced from the basic facts on equivariant homology of involution, see, e. g., Bre-72, Chapter 3.
Theorem 3.2.A follows from 3.2.B, since
and
Smith - Floyd Theorem can be applied to high-dimensional situation, too.
See Sections 5.3 and LABEL:s15.1d. In the one-dimensional case,
which we deal with here,
Theorem 3.2.B is easy to
prove without any homology tool, like the Smith theory. Namely, consider
the orbit space
These arguments contain more than just a proof of 2.3.A. In particular, they imply that
3.2.C .
In the case of an M-curve (i.e., if
Similarly, in the case of an
If
Note that
3.2.D .
The orbit space
Proof.
Assume that
On the other hand, if
3.2.E (Cf. 2.6.C).
If the curve is of type I, then
Proof.
This theorem follows from 3.2.C and the calculation of
the Euler characteristic of
3.2.F (Cf. 2.6.B).
Any M-curve is of type I.
Proof.
By 3.2.C, in the case of M-curve the orbit space
Now consider the simplest prohibition involving the placement of the real part of the flexible curve in the projective plane.
3.2.G .
The real part of a flexible curve is one-sided if and only if the degree is odd.
Proof.
The proof of 3.2.G coincides
basically with the proof of the same statement for algebraic curves.
One has to consider a real projective line transversal to the flexible
curve and calculate the intersection number of the complexification of
this line and the lfexible curve. On one hand, it is equal to the degree
of the flexible curve. On the other hand, the intersection points in
Rokhlin’s complex orientation formula also comes from topology. The proof presented in Section 2.7 works for a flexible curve.
At this point I want to break a textbook style exposition. Escaping a detailed exposition of prohibitions, I switch to a survey.
In the next two sections, the current state of prohibitions on the topology of a flexible curve of a given degree is outlined. (Recall that all formulations of this sort are automatically valid for real projective algebraic plane curves of the same degree.) After the survey a light outline of some proofs is proposed. It is included just to convey a general impression, rather than for more serious purposes. For complete proofs, see the surveys Wil-78, Rok-78, Arn-79, Kha-78, Kha-86, Vir-86 and the papers cited there.
3.3. A Survey of Prohibitions on the Real Schemes Which Come from Topology
In this section I list all
prohibitions on the real scheme of a flexible curve of degree
3.3.A .
A curve is one-sided if and only if it has odd degree.
This fact was given before as a corollary of Bézout’s theorem (see Section 1.3) and proved for flexible curves in Section 3.2 (Theorem 3.2.G).
3.3.B Harnack’s Inequality.
The number of
components of the set of real points of a curve of degree
Harnack’s inequality is undoubtedly the best known and most important prohibition. It can also be deduced from Bézout’s theorem (cf. Section 1.3) and was proved for flexible curves in Section 3.2 (Theorem 3.2.A).
In prohibitions 3.3.C–3.3.P the degree
Extremal Properties of Harnack’s Inequality
3.3.C Gudkov-Rokhlin Congruence.
In the case of an M-curve (i.e., if
3.3.D Gudkov-Krakhnov-Kharlamov Congruence.
In the case of an
The Euler characteristic of a component of the complement of a curve in
3.3.E Fiedler’s Congruence.
If the curve is an
M-curve,
3.3.F Nikulin’s Congruence.
If the curve is an
M-curve,
| (3) |
|
|
||
| (4) |
|
|
where
3.3.G Nikulin’s Congruence.
If the curve is an
M-curve,
Denote the number of even ovals with positive characteristic
by
Refined Petrovsky Inequalities
3.3.H .
3.3.I .
Refined Arnold Inequalities
3.3.J .
3.3.K .
Extremal Properties of the Refined Arnold Inequalities
3.3.L .
If
3.3.M .
If
Viro-Zvonilov Inequalities
Besides Harnack’s inequality, we know only one family of prohibition coming from topology which extends to real schemes of both even and odd degree. For proofs see VZ-92.
3.3.N Bound of the Number of Hyperbolic Ovals.
The number of
components of the complement of a curve of odd degree
The latter inequality also holds true for even
3.3.O Bound of the Number of Nonempty Ovals.
If
If
3.3.P Extremal Property of the Viro-Zvonilov Inequality.
If
where
3.4. Survey of Prohibitions on the Complex Schemes Which Come From Topology
Recall that
3.4.A See 2.6.A.
A curve with empty real point set is of type II.
3.4.B (See 2.6.C).
If the curve is of type I, then
3.4.C Rokhlin Complex Orientation Formula (see 2.7.C).
Let
Extremal Properties of Harnack’s Inequality
3.4.D (Cf. 2.6.B).
Any M-curve is of type I.
3.4.E Kharlamov-Marin Congruence.
Any
is of type I.
Extremal Properties of the Refined Arnold Inequalities
3.4.F .
If
3.4.G .
If
Extremal Properties of the Viro-Zvonilov Inequality
3.4.H .
Under the hypothesis of 3.3.P, the curve is of type I.
Congruences
3.4.I Nikulin-Fiedler Congruence.
If
The next two congruences are included violating a general promise given at the beginning of the previous section. There I promised exclude prohibitions which follow from other prohibitions given here. The following two congruences are consequences of Rokhlin’s formula 3.4.C. The first of them was discovered long before 3.4.C. The second was overlooked by Rokhlin in Rok-74, where he even mistakenly proved that such a result cannot exist. Namely, Rokhlin proved that the complex orientation formula does not imply any result which would not follow from the prohibitions known by that time and could be formulated solely in terms of the real scheme. Slepian congruence 3.4.K for M-curves is the only counter-example to this Rokhlin’s statement. Slepian was Rokhlin’s student, he discovered a gap in Rokhlin’s arguments and deduced 3.4.K.
3.4.J Arnold Congruence (see 2.7.D).
If
3.4.K Slepian Congruence.
If
Rokhlin Inequalities
Denote by
3.4.L .
If the curve is of type I and
3.4.M .
If the curve is of type I and
3.5. Ideas of Some Proofs
Theorems formulated in 3.3 and 3.4 are very different in their profundity. The simplest of them were considered in Subsection 3.2.
Congruences
There are two different approaches to proving congruences. The first is
due basically to Arnold Arn-71 and Rokhlin Rok-72. It is based on
consideration of the intersection form of two-fold covering
The second approach is due to Marin Mar-80. It is based on
application of the Rokhlin-Guillou-Marin congruence modulo 16
on characteristic surface in a 4-manifold, see GM-77.
It is applied either to the surface in the quotient space
The first approach was applied also in high-dimensional situations. The second approach worked better than the second one for curves on surfaces distinct from projective plane, see Mik-94. Both were used for singular curves KV-88.
Inequalities
Inequalities 3.3.H, 3.3.I, 3.3.J, 3.3.K,
3.4.J and 3.4.K are proved along the same scheme,
originated by Arnold Arn-71. One constructs an auxiliary
manifold, which is the two-fold covering of
3.6. Flexible Curves of Degrees ≤5
In this
subsection, I show that for degrees
Degrees
Degree 4. By the Arnold inequlity 3.3.K, a flexible curve
of degree 4 cannot have a nest of depth 3. By the Arnold inequality
3.3.J, it has at most one nonempty positive oval, and if it has a
nonempty oval then, by the extremal property 3.3.L of this
inequality, the real scheme is
From the Klein congruence 3.4.B, it follows that the real schemes
Degree 5. By the Viro-Zvonilov inequality 3.3.O, a
flexible curve of degree 5 can have at most one nonempty oval. By the
extremal property of this inequality 3.3.P, if a flexible curve
of degree 5 has a nonempty oval, then its real scheme is
From the Klein congruence 3.4.B, it follows that the real schemes
3.7. Sharpness of the Inequalities
The arsenal of constructions in Section 1 and the supply of curves constructed there, which are very modest from the point of view of classification problems, turn out to be quite rich if we are interested in the problem of sharpness of the inequalities in Section 3.3.
The Harnack curves of even degree
which were constructed in Section 1.6 (see also Section 1.9) not only show that Harnack’s inequality 3.3.B is the best possible, but also show the same for the refined Petrovsky inequality 3.3.H.
One of the simplest variants of Hilbert’s construction (see Section
1.10) leads to the construction of a series of M-curves of
degree
The refined Arnold inequality 3.3.J is best possible for any
even
The last construction (doubling), if applied to an M-curve of odd degree,
shows that the refined Arnold inequality 3.3.K is the best possible
for
3.8. Prohibitions not Proven for Flexible Curves
In conclusion of this section, let us come back to algebraic curves. We see that to a great extent the topology of their real point sets is determined by the properties which were included into the definition of flexible curves. In fact, it has not been proved that it is not determined by these properties completely. However some known prohibitions on topology of real algebraic curves have not been deduced from them.
As a rule, these prohibitions are hard to summarize, in the sense that it is difficult to state in full generality the results obtained by some particular method. To one extent or another, all of them are consequences of Bézout’s theorem.
Consider first the restrictions which follow directly from the Bézout
theorem. To state them, we introduce the
following notations. Denote by
3.8.A .
3.8.B .
These statements suggest a whole series of similar assertions.
Denote by
3.8.C (Generalization of Theorem 3.8.A).
If
In particular, if
3.8.D (Generalization of Theorem 3.8.B).
If
In particular, if
The following two restrictions on complex schemes are similar to Theorems 3.8.A and 3.8.B. However, I do not know the corresponding analogues of 3.8.C and 3.8.D.
3.8.E .
If
3.8.F .
If
Here I will not even try to discuss the most general prohibitions which do not come from topology. I will only give some statements of results which have been obtained for curves of small degree.
3.8.G .
There is no curve of degree 7 with
the real scheme
3.8.H .
If an M-curve of degree 8 has
real scheme
3.8.I .
If an
Proofs of 3.8.G and 3.8.H are based on technique initiated by Fiedler Fie-82. It will be developed in the next Section.
4. The Comlexification of a Curve from a Real Viewpoint
In the previous two sections we discovered that a knowledge on topology of the complexification gives restriction on topology of real part of the curve under consideration. More detailed topological information on complexification can be obtained using geometric constructions involving auxiliary curves. They use Bézout theorem. Therefore they cannot be applied to flexible curves. Here we consider first the simplest of arguments of that sort, and then obtain some special results on curves of low degrees (up to 8) which, together with forthcoming constructions will be useful in solution of some classification problems.
We will use the simplest auxiliary curves: lines. Consideration of a pencil of lines (the set of all lines passing through a point) and intersection of a curve with lines of this pencil can be thought of as a study of the curve by looking at it from the common point of the lines. However, since imaginary lines of the pencil can be included into this study and even real lines may intersect the curve in imaginary points, we have a chance to find out something on the complex part of the curve.
4.1. Curves with Maximal Nest Revised
To begin with, I present another proof of Theorem 2.6.D. It gives slightly more: not only that a curve with maximal nest has type I, but that its complex orientation is unique. This is not difficult to obtain from the complex orientation formula. The real cause for including this proof is that it is the simplest application of the technique, which will work in this section in more complicated situations. Another reason: I like it.
4.1.A .
If a nonsingular real plane projective curve
Recall that by Corollary 1.3.C of the Bézout theorem a nest of
a curve of degree
for even
if
Proof of 4.1.A.
Take a point
The real part
The projection
4.2. Fiedler’s Alternation of Orientations
Consider the pencil of real lines passing through the intersection
point of real lines
A point of tangency of two oriented curves is said to be positive if the orientations of the curves define the same orientation of the common tangent line at the point, and negative otherwise.
The following theorem is a special case of the main theorem of Fiedler’s paper Fie-82.
4.2.A Fiedler’s Theorem.
Let
I give here a proof, which is less general than Fiedler’s original one. I hope though that it is more visualizable.
Roughly speaking, the main idea of this proof is that, looking at a curve, it is useful to move slightly the viewpoint. When one looks at the intersection of the complexification of a real curve with complexification of real lines of some pencil, besides the real part of the curve only some arcs are observable. These arcs connect ovals of the curve, but they do not allow to realize behavior of the complexification around. However, when the veiwpoint (= the center of the pencil) is moving, the arcs are moving too sweeping ribbons in the complexification. The ribbons bear orientation inherited from the complexification and thereby they allow to trace relation between the induced orientation of the ovals connected by the arcs. See Figure 26
Proof of 4.2.A.
The whole situation described in
the 4.2.A is stable under small moves of the point
Choose a point
Thus the arcs
The next thing to do is to obtain prohibitions on complex schemes using Fiedler’s theorem. It takes some efforts because we want to deduce topological results from a geometric theorem. In the theorem it is crucial how the curve is positioned with respect to lines, while in any theorem on topology of a real algebraic curve, the hypothesis can imply some particular position with respect to lines only implicitely.
Let
Fix a complex orientations of
A point of tangency of
If
4.2.B [Fie-82, Lemma 2] .
Let
Denote the closure of
4.2.C .
The signs of ovals with respect to
The next theorem follows in an obvious way from 4.2.C. Contrary to the previous one, it deals with the signs of ovals with respect to the one-sided component in the case of odd degree and outer ovals in the case of even degree.
4.2.D [Fie-82, Theorem 3].
If the degree of a curve
then the
signs of these ovals alternate. If degree of
4.3. Complex Orientations and Pencils of Lines. Alternative Approach
In proofs of 3.8.G, 3.8.H and 3.8.I, the theory developed in the previous section can be replaced by the following Theorem 4.3.A. Although this theorem can be obtained as a corollary of Theorem 4.2.C, it is derived here from Theorem 2.3.A and the complex orientation formula, and in the proof no chain of ovals is used. The idea of this approach to Fiedler’s alternation of orientations is due to V. A. Rokhlin.
4.3.A .
Let
Proof.
Assume the contrary: suppose that with respect to a
complex orientation of
4.4. Curves of Degree 7
In this section Theorem 3.8.G is
proved, i.e. it is proved that there is no nonsingular curve of degree
7 with real scheme
Assume the contrary: suppose that there exists a nonsingular
curve
Being an M-curve,
4.4.A Lemma.
Proof.
Let
It is clear that
Therefore,
and, since
The next ingredient in the proof of Theorem 3.8.G is a kind of convexity in disposition of interior ovals. Although we study a projective problem, it is possible to speak about convexity, if it is applied to interior ovals. The exact sense of this convexity is provided in the following statement.
4.4.B Lemma.
Let
Proof.
A line intersecting two interior ovals cannot intersect any other interior oval. Furthermore, it intersects each of these two interior ovals in two points, meets the nonempty oval in two points and the one-sided component in one point. (This follows from the following elementary arguments: the line intersects the one-sided component with odd multiplicity, it has to intersect the nonempty oval, since it intersects ovals inside of it, it can intersect any oval with even multiplicty and by Bézout theorem the total number of intersection points is at most 7.) The real point set of the line is divided by the intersection points with the nonempty oval into two segments. One of these segments contains the intersection point with the one-sided component, the other one is inside the nonempty oval and contains the intersections with the interior ovals. A smaller segment connects the interior ovals inside the nonempty ovals. Thus any points inside two interior ovals can be connected by a segment of a line inside the exterior nonemty oval. See Figure 4.4.
Choose a point inside each interior oval and connect these points by
segments inside the exterior oval. If the lines guaranteed by
4.4.B exist, then the segments comprise a convex polygon.
Otherwise, there exist interior ovals
To prove that this is impossible, assume that this is the case and
construct a conic
Now let us estimate the number of intersection points of the conic and
the original curve
End of Proof of Theorem 3.8.G.
Assume that a curve
5. Introduction to Topological Study of Real Algebraic Spatial Surfaces
5.1. Basic Definitions and Problems
Our consideration of real algebraic surfaces will be based on
definitions similar to the definitions that we used in the case of
curves. In particular, by a real algebraic surface of degree
Obvious changes adapt definitions of sets of real and
complex points, singular points, singular and nonsingular curves and
rigid isotopy to the case of surfaces in
5.1.A Topological Classification Problem.
Up to homeomorphism, what are the possible sets of real points of a
nonsingular real projective algebraic surface of degree
However, the isotopy classification problem 1.1.B splits into two problems:
5.1.B Ambient Topological Classification Problem.
Classify
up to homeomorphism the pairs
5.1.C Isotopy Classification Problem.
Up to ambient isotopy, what are the possible sets of real points of a
nonsingular a nonsingular real projective algebraic surface of degree
The reason for this splitting is that, contrary to the case of
projective plane, there exists a homeomorphism of
5.1.D Amphichirality Problem.
Which nonsingular real
algebraic surfaces of degree
Each of these problems has been solved only for
and the union of one-sheeted hyperboloid and an imaginary quadric (perturbed, if you wish to have a surface without singular points even in the complex domain)
Similar splitting happens with the rigid isotopy classification problem. Certainly, it may be transferred literally:
5.1.E Rigid Isotopy Classification Problem.
Classify up to rigid isotopy the nonsingular surfaces of degree
However, since there exists a projective transformation of
5.1.F Rough Projective Classification Problem.
Classify up to rigid isotopy and projective transformation the nonsingular
surfaces of degree
Again, as in the case of topological isotopy and homeomorphism problem, the difference between these two problems is an amphichirality problem:
5.1.G Rigid Amphichirality Problem.
Which
nonsingular real algebraicsurfaces of degree
Problems 5.1.E, 5.1.F and 5.1.G have been solved
also for
5.2. Digression: Topology of Closed Two-Dimensional
Submanifolds of ℝP3
For brevity, we shall refer to
closed two-dimensional submanifolds of
Since the homology group
In the first case it divides the projective space into two domains being the boundary for both domains. Hence, the surface divides its tubular neighborhood, i. e. it is two-sided.
In the second case the complement of the surface in the projective space is connected. (If it was not connected, the surface would bound and thereby realize the zero homology class.) Moreover, it is one-sided.
The latter can be proved in many ways. For example, if the surface was
two-sided and its complement was connected, there would exist a
nontrivial infinite cyclic covering of
Another proof: take a projective plane, make it transversal to the
surface, and consider the curve which is their intersection. Its
homology class in
A connected surface two-sidedly embedded in
A one-sidedly embedded surface is nonorientable. Indeed, its normal
bundle is nonorientable, while the restriction of the tangent bundle
of
Contrary to the case of two-sided surfaces, in the case of one-sided surfaces there is an additional restriction on their topological types.
5.2.A .
The Euler characteristic of a connected surface one-sidedly embedded to
In particular, it is impossible to embed a Klein
bottle to
Proof of 5.2.A.
Let
Consider the disjoint sum
A one-sided connected surface in
A two-sided connected surface in
If any loop on a connected surface
It may happen, however, that there is no isotopy relating the embedding
of a contractible surface with a map to an affine part of
As it was stated above, the complement
The simplest example of this situation is provided by a one-sheeted
hyperboloid. It is homeomorphic to torus and its complement consists of
two solid tori. So, this is a Heegaard decomposition of
A connected surface decomposing
If a connected surface
A contractible connected surface
At most one component of a (closed) surface embedded in
Moreover, if an embedded surface has a one-sided component, then
all other components are contractible. The contractible components are
naturally ordered: a contractible component of a surface can contain
other contractible component in its interior and this gives rise to a
partial order in the set of contractible components. If the interior of
contractible surface
The connected components of a surface embedded in
Consider now a (closed) surface without one-sided components. It may
contain several noncontractible components. They decompose the
projective space into connected domains, each of which is not
contractible in
Contractible components of the surface are distributed in the domains.
Contractible components which are contained in different domains cannot
envelope one another. Contractible components of the surface which lie
in the same domain are partially ordered by enveloping. They divide
the domain into regions. Each domain contains only one region which is
not contractible in
The region tree of a surface contains a subtree isomorphic to the domain tree, since one can assign to each domain the unique noncontractible region contained in the domain and two domains are adjacent iff the noncontractible regions contained in them are adjacent. The complement of the noncontractible domains tree is a union of adjacency trees for contractible subdomains contained in each of the domains.
Let us summarize what can be said about topology of a spatial surface in the terms described above.
If a surface is one-sided (i. e., contains a one-sided component), then it is a disjoint sum of a projective plane with handles and several (maybe none) spheres with handles. Thus, it is homeomorphic to
where
All two-sided components are contractible and ordered by
enveloping. The order is easy to incorporate into the notation of the
topological type above. Namely, place notations for components
enveloped by a component
denotes a surface consisting of a
projective plane, two tori, which do not envelope any other
component, a sphere, which envelopes a torus and a sphere without
components inside them and a two spheres with two handles
each of which envelopes empty sphere and torus. To make the notations
shorter, let us agree to skip index 0, i. e. denote projective plane
If a surface is two-sided (i. e. does not contain a one-sided
component), then it is a disjoint sum
In these notations,
denotes a two-sided surface containing three noncontractible components. One of them is a torus, two others are spheres with two handles. The torus bounds a domain containing a contractible empty torus and a sphere enveloping three empty spheres. There is a domain bounded by all three noncontractible components. It contains a contractible empty sphere with three handles. Each of the noncontractible spheres with two handles bounds a domain containing empty contractible torus and three empty spheres.
This notation system is similar to notations used above to described isotopy types of curves in the projective plane. However, there is a fundamental difference: the notations for curves describe the isotopy type of a curve completely, while the notations for surfaces are far from being complete in this sense. Although topological type of the surface is described, knotting and linking of handles are completely ignored. In the case when there is no handle, the notation above does provide a complete description of isotopy type.
5.3. Restrictions on Topology of Real Algebraic Surfaces
As in the case of real plane projective curves,
the set of real points of a nonsingular spatial surface of degree
There are some other restrictions on topology of a nonsingular surface
of degree
5.3.A On Number of Cubic’s Components.
The set of real points of a nonsingular surface of degree three consists of at most two components.
Proof.
Assume that there are at least three components. Only one of
them is one-sided, the other two are contractible. Connect with a line
two contractible components. Since they are zero-homologous, the line
should intersect each of them with even intersection number. Therefore
the total number of intersection points (counted with multiplicities)
of the line and the surface is at least four. This contradicts to the
Bézout theorem, according to which it should be at most
three.
5.3.B On Two-Component Cubics.
If the set of real
points of a nonsingular surface of degree 3 consists of two components,
then the components are homeomorphic to the sphere and projective plane
(i. e., this is
Proof.
Choose a point inside the contractible component. Any line
passing through this point intersects the contractible component at
least in two points. These points are geometrically distinct, since the
line should intersect also the one-sided component. On the other hand,
the total number of intersection points is at most three according to
the Bézout theorem. Therefore any line passing through the selected
point intersects one-sided component exactly in one point and two-sided
component exactly in two points. The set of all real lines
passing through the point is
5.3.C Estimate for Diameter of Region Tree.
The
diameter of the region tree66Here by the diameter of a tree
it is understood the maximal number of edges in a simple chain of edges
of the tree, i. e., the diameter of the tree in the internal metric,
with respect to which each edge has length 1. of a nonsingular surface
of degree
Proof.
Choose two vertices of the region tree the most distant from
each other. Choose a point in each of the coresponding regions
and connect the points by a line.
5.3.D .
The set of real points of a nonsingular surface of degree 4 has at most two noncontractible components. If the number of noncontractible components is 2, then there is no other component.
Proof.
First, assume that there are at least three noncontractible components. Consider the complement of the union of three noncontractible components. It consists of three domains, and at least two of them are not adjacent (cf. the previous subsection: the graph of adjacency of the domains should be a tree). Connect points of nonadjacent domains with a line. It has to intersect each of the three noncontractible components. Since they are zero-homologous, it intersects each of them at least in two points. Thus, the total number intersection points is at least 6, which contradicts to the Bézout theorem.
Now assume that there are two noncontractible components and some
contractible component. Choose a point
Otherwise (i. e. if the contractible component lies in the domain
adjacent to both noncontractible components), choose inside each of the
two other domains an embedded circle, which does not bound in
5.3.E Remark.
In fact, if a nonsingular quartic surface has two noncontractible components then each of them is homeomorphic to torus. It follows from an extremal property of the refined Arnold inequality 5.3.L. I do not know, if it can be deduced from the Bézout theorem. However, if to assume that one can draw lines in the domains of the complement which are not adjacent to both components, then it is not difficult to find homeomorphisms between the components of the surface and the torus, which is the product of these two lines. Cf. the proof of 5.3.B.
5.3.F Generalization of 5.3.D.
Let
The proof is a straightforward generalization of the proof of
5.3.D.
Surprisingly, Bézout theorem gave much less restrictions in the case of surfaces than in the case of plane curves. In particular, it does not give anything like Harnack Inequality. Most of restrictions on topology of surfaces are analogous to the restrictions on flexible curves and were obtained using the same topological tools. Here is a list of the restrictions, though it is non-complete in any sense.
The restrictions are formulated below for a nonsingular real algebraic
surface
5.3.G Generalized Harnack Inequality.
5.3.H Remark.
This is a special case of Smith-Floyd Theorem
3.2.B, which in the case of curves implies Harnack
Inequality, see Subsections 3.2. It says that
for any involution
Applying this to the complex conjugation involution of the
complexification
5.3.I Extremal Congruences of Generalized Harnack Inequality.
If
then
If
5.3.J Petrovsky - Oleinik Inequalities.
Denote the numbers of orientable components of
5.3.K Refined Petrovsky - Oleinik Inequality.
If
5.3.L Refined Arnold Inequality.
Either
or
5.4. Surfaces of Low Degree
Surfaces of degree 1
and 2 are well-known. Any surface of degree 1 is a projective plane.
All of them are transformed to each other by a rigid isotopy
consisting of projective transformations of the whole ambient space
Nonsingular surfaces of degree 2 (nonsingular quadrics) are of three types. It follows from the well-known classification of real nondegenerate quadratic forms in 4 variables up to linear transformation. Indeed, by this classification any such a form can be turned to one of the following:
-
(1)
+x20+x21+x22+x23 , -
(2)
+x20+x21+x22−x23 , -
(3)
+x20+x21−x22−x23 , -
(4)
+x20−x21−x22−x23 , -
(5)
−x20−x21−x22−x23 .
Multiplication by
The first of the types consists of quadrics with empty set of real
points. In traditional analytic geometry these quadrics are called
imaginary ellipsoids. A canonical representative of this class is
defined by equation
The second type consists of quadrics with the set of real points
homeomorphic to sphere. In the notations of the previous section this
is
The third type consists of quadrics with the set of real points
homeomorphic to torus. They are known as one-sheeted hyperboloids. The
set of real points is not contractible (it contains a line), so in the
notations above it should be presented as
Quadrics of the last two types (i. e., quadrics with nonempty real part)
can be obtained by small perturbations of a union of two real planes.
To obtain a quadric with real part homeomorphic to sphere, one
may perturb the union of two real planes in the following way. Let the
plane be defined by equations
with a small
To obtain a noncontractible nonsingular quadric (one-sheeted
hyperboloid), one can perturb the same equation
Nonsingular surfaces of degree 3 (nonsingular cubics) are of five types. Here is the complete list of there topological types:
Let us prove, first, that only topological types from this list can be
realized. Since the degree is odd, a nonsingular surface has to be
one-sided. By 5.3.D if it is not connected, then it is
homeomorphic to
All the five topological types are realized by small perturbations of unions of a nonsingular quadric and a plane transversal to one another. This is similar to the perturbations considered above, in the case of spatial quadrics. See Figures 35 and 36.
An alternative way to construct nonsingular surfaces of degree 3 of all the topological types is provided by a connection between nonsingular spatial cubics and plane nonsingular quartics. More precisely, there is a correspondence assigning a plane nonsingular quartic with a selected real double tangent line to a nonsingular spatial cubic with a selected real point on it. It goes as follows. Consider the projection of the cubic from a point selected on it to a plane. The projection is similar to the well-known stereographic projection of a sphere to plane.
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